Problems

a) There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?

b) The same question, if there are 20 coins, but you are allowed to turn over 19.

Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?

Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?

A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

One of the four angles formed when two straight lines intersect is \(41^{\circ}\). What are the other three angles equal to?

A wide variety of questions in mathematics starts with the question ’Is it possible...?’. In such problems you would either present an example, in case the described situation is possible, or rigorously prove that the situation is impossible, with the help of counterexample or by any other means. Sometimes the border between what seems should be possible and impossible is not immediately obvious, therefore you have to be cautious and verify that your example (or counterexample) satisfies the conditions stated in the problem. When you are asked the question whether something is possible or not and you suspect it is actually possible, it is always useful to ask more questions to gather additional information to narrow the possible answers. You can ask for example "How is it possible"? Or "\(\bf Which\) properties should the correct construction satisfy"?

Cut a square into two equal:
1. Triangles.
2. Pentagons
3. Hexagons.

Cut the following figure into two equal parts.

Find all rectangles that can be cut into \(13\) equal squares.